Theorem: Flatness of the constant affine representative

Translate any constant representative $c{\eta }_D$, $c>0$, across the affine carrier. Its Levi-Civita connection coefficients vanish in affine coordinates, and its Riemann, Ricci, and Einstein tensors are zero.

Likewise, a one-parameter family of spatially constant representatives has zero curvature at every parameter value. Its first metric variation has zero linearized Einstein tensor.

Thus a matching of this unchanged constant geometry to a nonzero source cannot be repaired by choosing another positive constant normalization.

Proof. For constant metric components,

The curvature constructed from derivatives and products of these coefficients is zero. Constant rescaling leaves the components constant.

In a parameter family $g_{\lambda }$ of constant forms, all spacetime derivatives remain zero for every $\lambda$. The variation $h={{\partial }_{\lambda }{g}_{\lambda }|}_0$ is also spatially constant, so every second-derivative term in the linearized Einstein operator vanishes. A parameter derivative is not a spacetime derivative.